1. Field of the Invention
This invention relates generally to computer curve construction systems and methods. In particular, it describes additional embodiments of a “Computer Curve Construction System” described in U.S. Pat. No. 6,441,823, Ananya, where the chosen curves are conical sections.
2. Description of Related Art
A conic is a mathematically defined curve. For example, as shown in FIG. 1, typically a conic it is drawn (constructed) with a computer curve drawing program by setting a start point a0, an end point a1, and a control point a2, and then choosing any arbitrary point a3 on the conic. The tangent directions of the conic are determined by the control point a2, and the shape of the conic is determined by the chosen arbitrary point a3. However, simply choosing an arbitrary point a3 that will be somewhere on the conic does not allow a program operator to intuitively anticipate the shape of the conic that will be constructed by the computer.
In contrast, U.S. Pat. No. 6,441,823, Ananya, B., describes a computer curve construction system for creating peak-point curves by selecting a star point, a start tangent direction, an end point spaced from the start point, an end tangent direction, and a peak point somewhere between the start and end points where the program calculates and draws a curve connecting the start and end points passing through the peak point where the tangent of the curve at the peak point is parallel to the chord between the start and end points. [See claims 13, 14, 15, & 16, Col. 24 ll 7–54.] By defining a peak point, the curve construction system described in Ananya enables the operator to better anticipate, intuitively, the shape of the curve the program will construct responsive to operator input.
Conical section or conics have the advantage of being generally more geometric than Bezier curves. This geometric feature allows for easier operator conceptualization of a contemplated curve before inputting defining parameters. Secondly, behavior of conics are computationally predictable with established relationships that exist or can be easily defined. Yet for formulations and accurate numeric approximations by a computer (CPU) constructing the conic, the defining signature constraints on the operator selected parameters must satisfy particular conditions before the computer program can allow construction of a desired curve.